1. Topic 18: Cartilage Mechanics
Types, Locations, Mechanical Functions
FUNCTIONAL COMPONENTS OF CARTILAGE
DISEASES INVOLVING CARTILAGE
FRICTION, LUBRICATION AND WEAR
TENSILE PROPERTIES
SHEAR PROPERTIES
16-Feb-1999

2. Cartilage Types & Locations
hyaline cartilage (hylos = glassy)
costal, nasal, tracheo-bronchial, and articular cartilage; cartilage model (pre-skeletal)
white fibrocartilage
intervertebral disc, articular disc (knee, wrist, jaw), bony grooves lodging tendons
yellow elastic fibrocartilage
external ears, larynx, epiglottis, arytenoids

3. Prenatal development of long bone
A, the mesenchymal model, 5 weeks
B, the cartilage model, 6 weeks
C and D, the beginning of ossification, 7 to 12 weeks

4. Growth of Long Bone
Later stages of prenatal development of a long bone. A, 6 months. B, at birth
Growth of a long bone (the femur) during childhood. In freely movable joints the ends of the bones are capped with articular cartilage. The epiphyseal cartilage plate provides for the longitudinal growth of the bone.

5. Mechanical Functions of Articular Cartilage
Joint motion with minimal friction or wear, lower extremities undergo ~1 million loading cycles/year
distribute and transmit stress to bone

6. Functional Components of Cartilage
Two major phases - solid and fluid (“biphasic”)

7. Components of Cartilage: Cells
(= chondrocytes)
occupy only 1-10% of the volume of cartilage (young cartilage has a higher density and volume fraction of cells than adult cartilage)
 in mechanical measurement and analysis, contribution of chondrocytes usually neglected.
Functions:
synthesize the extracellular matrix (ECM)
synthesize enzymes that degrade ECM
respond to changing functional biomechanical demands

8. Functional Components of Cartilage
Light micrograph of mature rabbit articular cartilage. S, tangential zone; T, transitional zone; UR, upper radial zone; LR, lower radial zone; C, calcified cartilage; B, subchondral bone plate.
Electron micrographs of chondrocytes originating from the different zones of mature articular cartilage.
A: Tangential zone chondrocyte;
B: Transitional zone chondrocyte;
C, D: Upper (C) and lower (D) radial zone chondrocytes.

9. Extracellular Matrix: Solid Phase
Collagen
~60% of dry weight of cartilage
mostly type II collagen
(a chain  triple helical molecule  fibril  fiber)
zonal collagen architecture (superficial tangential, middle, deep)
functions:
tensile stiffness and strength
mesh that immobilizes proteoglycan aggregates
Collagen fibril structure

10. Cartilage Collagen Network
Layered structure of collagen network showing three distinct regions (A); corresponding SEM collagen fibrillar arrangement (B).

11. Components of Cartilage: Proteoglycan
~30% of the dry weight of cartilage
terminology:
proteoglycan aggregate = proteoglycan monomers, aggregated onto a hyaluronate backbone, in an interaction stabilized by link proteins
proteoglycan monomer - protein core to which are covalently attached ~ sugar polymers (glycosaminoglycans)
glycosaminoglycans are highly charged (acidic) molecules due to -SO3- & -COO- that are negatively ionized at physiologic pH
charge density of articular cartilage is high (negative mol/L tissue fluid)

12. Proteoglycans (cont'd)
electrostatic repulsion of neighboring GAG polymers makes proteoglycans tend to swell and resist compression
electrostatic attraction and concentration of positive counter-ions (Na+, H+), and repulsion and dilution of negative co-ions (Cl-); e.g., for a bathing medium a concentration of NaCl of 0.11 M and a charge density of -0.2 mol/L, the extracellular matrix concentration [Na+] = 0.26 M and [Cl-] = 0.05 M
functions:
compressive stiffness
modulates ionic milieu of extracellular fluid

13. Functional Components of Cartilage
Extracellular Fluid Phase
(water, ions: see above)
A. Schematic depiction of an aggregating proteoglycan monomer composed of keratin sulfate and chondroitin sulfate chains bound covalently to a protein core molecule.
B. A representation of a proteoglycan aggregate which is composed of monomers noncovalently attached to hyaluronic acid with stabilizing link proteins.

14. Schematic Summary: Joint Loading
A. Chondrocytes synthesize collagens, proteoglycans, hyaluronate, and link protein. These components assemble into a stable extracellular matrix; subsequent catabolic processes cause the release of certain components from the tissue. Alternatively, some biosynthetic products may never become incorporated into the matrix. These metabolic processes may be influenced by mechanical, chemical, or electrical phenomena.

15. Schematic of Joint Loading
B. Physical phenomena: chondrocyte deformation, hydrostatic pressurization, fluid flow, electric fields (streaming potentials and currents), matrix consolidation, and physico-chemical alterations (altered ion concentrations and osmotic pressure).

16. Compression of Cartilage

17. Diseases of Cartilage: Osteoarthritis
“degenerative joint disease,” a wear and tear phenomenon
gross changes:
erosion of the joint surface
reactive bone formation
microscopic changes:
fissures in the articular surface
cell division
increased incidence with aging, obesity, joint injury
extremely common (80% of Americans > age 65 have radiological evidence)
treatment:
analgesics
 total joint replacement
potential cause: (see figure)
excessive joint loading
matrix degradation process unknown

18. Diseases Involving Cartilage

19. Diseases Involving Cartilage
Knee joint opened anteriorly reveals large erosion of articular cartilages of femur and patella with cartilaginous excrescences at intercondylar notch.
Section of articular cartilage shows fraying of surface and deep cleft. Hyaline cartilage abnormal with clumping of chondrocytes.

20. Diseases Involving Cartilage
Hypothesis of the etiopathogenesis of osteoarthrosis due to excessive joint loading. (I) Normal articular cartilage. Dots represent proteoglycans, curved lines represent collagens, and oval structures represent chondrocytes. (ii) Proteoglycan loss exposes the superficial collagen fibrils; condition is still reversible. (iii) Injury of collagen fibrils in the cartilage surface; additional loss of proteoglycans; “point of no return.”

21. Diseases of Cartilage: Rheumatoid Arthritis
“autoimmune” disease
gross changes:’
synovial thickening due to inflammation
pannus formation
erosive destruction of cartilage and bone
microscopic changes:
synovial cell proliferation
cartilage destruction most severe under pannus
familial (“inherited”)
much less common than osteoarthritis
treatment of flares:
immunosuppressive drugs
-> total joint replacement
potential cause: (see figure)
initiating/triggering factor unknown
inflammatory process and enzymes
stimulation of chondrocyte-mediated matrix degradation

22. Diseases Involving Cartilage
Progressive stages in joint pathology.
1. Acute inflammation of synovial membrane (synovitis) and beginning proliferative changes.
2. Progression of inflammation with pannus formation; beginning destruction of cartilage and mild osteoporosis.
3. Subsidence of inflammation; fibrous ankylosis.
4. Bony ankylosis; advanced osteoporosis.

23. Diseases Involving Cartilage
Section of proximal interphalangeal joint shows marked destruction of both articular cartilages and subchondral bone; replacement by fibrous and granulation tissue, which has obliterated most of joint space and invaded bone.
Knee joint opened anteriorly, patella reflected downward. Thickened synovial membrane inflamed; polypoid outgrowths and numerous villi (pannus) extend over rough articular cartilages of femur and patella.
Section of synovial membrane shows villous proliferation with extensive lymphocytic and plasma cell infiltration and numerous blood vessels. Synovial lining cells are elongated and arranged in palisade formation.

24. Diseases Involving Cartilage
The chondrocyte has a primary role in joint cartilage matrix turnover.

25. Friction, Lubrication and Wear
Synovial Fluid
Synovial transport pathways (not drawn to scale)

26. Coefficient of Friction

27. Friction, Lubrication and Wear
A joint friction experiment using an isolated animal synovial joint.
Schematic of typical test configuration

28. Coefficient of Friction

29. Friction, Lubrication and Wear
A. A simple pendulum device with the human hip joint as its fulcrum is used to measure the coefficient of friction between femoral head (shown) and the acetabulum (not shown) by the decay of the amplitude of the pendulum motion.

30. Friction, Lubrication and Wear
B. A typical set of curves for the coefficient of friction versus the number of cycles from one hip joint specimen under suddenly loaded conditions - unlubricated with synovial fluid - showing a longer period of swing and a lower coefficient of friction with increasing load.
C. Differences of the coefficient of friction between unlubricated and synovial fluid lubricated hip joints with varying applied loads. At a load >600 N, no differences in coefficients of friction were observed.

31. Explant Studies
Specimens cut from the bovine humeral head and glenoid surface provide conforming surfaces for studies on interfacial friction.
Variation of the coefficient of friction under various lubrication and loading conditions.

32. Dependence on Collagen Alignment & Fibrillation
Tensile Properties
Dependence on Collagen Alignment & Fibrillation
Typical test configuration
Typical stress-strain curve for articular cartilage in a uniaxial experiment

33. Tensile Properties Dependence on collagen fibrillation
Equilibrium tensile modulus (MPa) of human articular cartilage; dependence on depth and degeneration

34. Young’s Modulus: Cartilage cf. Tendon and Skin

35. Dependence on Collagen Content
Shear Properties
Dependence on Collagen Content
Typical test configuration
Shear test of a circular specimen
A schematic of cartilage in pure shear. The tension of collagen provided shear stiffness.

36. Shear Properties Dependence on collagen content
A direct correlation between the collagen content (by net weight) and magnitude of dynamic modulus |G*| for bovine articular cartilage. The compressive clamping strain is 20% and frequency f = 1 Hz.

37. Confined Compression & Swelling
Equilibrium confined compression modulus
(Above) Schematic of measurement of the bulk longitudinal modulus of a mechanically linear tissue equilibrated at NaCl concentration co.

38. Confined Compression & Swelling
Role of fixed charge density

39. Bulk Longitudinal Modulus
Bulk longitudinal modulus HA(c) for articular cartilage (b) as a function of NaCl concentration.

40. Confined Compression: Stress-relaxation
Schematic representation of fluid exudation and redistribution within cartilage during a rate-controlled, compression stress-relaxation experiment (lower left figure). The horizontal bars in the upper figures indicate the distribution of strain in the tissue. The lower graph (right) shows the stress response during compression phase (O, A, B) and the relaxation phase (B, C, D, E).

41. Confined Compression: Creep
Schematic representation of fluid exudation and redistribution within a poroelastic material such as cartilage during a step application of stress and the resulting creep deformation.

42. Confined Compression & Swelling

43. Confined Compression & Swelling
ASSUMPTIONS
(1) the cells do not contribute to the physical properties and can be ignored
(2) cartilage consists of homogeneous (spatially uniform) solid and fluid phases
(3) the solid matrix behaves as a linearly elastic and isotropic (idential in all directions) material
(4) there is frictional drag that is proportional to the fluid velocity (relative to the solid) and characterized by the hydraulic (open circuit) permeability
(5) the fluid and current flow are coupled to the hydrostatic pressure and electrical potential gradients through electrokinetic coupling coefficients that obey Onsager reciprocity
(6) the stresses and strains are infinitesimal

44. Poroelastic Modeling of Cartilage
constitutive law:
non-viscous fluid & Hookean elastic solid
A constitutive law for a poroelastic material composed of a Hookean elastic solid and a nonviscous fluid that relates the compressive stress (z,t) to the compressive strain (z,t), the equilibrium confined compression modulus Ha, and the fluid pressure Pf(z,t) is:
The uniaxial infinitesimal compressive strain within the tissue is related to the tissue displacement u(z,t) by:

45. Poroelastic Modeling of Cartilage
Conservation of mass
Continuity (mass balance) for a poroelastic material (composed of incompressible solid and fluid phases) relates the relative fluid velocity U(z,t) to the strain rate by:
Conservation of momentum
Conservation of momentum in the frequency range in which inertia is negligible is:

46. Darcy’s law: hydraulic permeability
The fluid-solid interaction is described by the phenomenological relations of non-equilibrium thermodynamics:
where the electrokinetic coupling coefficients are equal (k12 = k21) by Onsager reciprocity and k22 is the electrical conductance.
The open circuit, Darcy hydraulic permeability is:

47. Poroelastic Modeling of Cartilage
Conservation of current
Conservation of current requires that:
the equation of motion: a diffusion equation!
Equations (1-5) can be solved to yield an equation of motion of the displacement, which takes the form of a diffusion equation:
where we have used the definition of hydraulic permeability (6) and assumed that the current at the boundaries (i.e., an open-circuit experimental configuration) is 0, and then extended this constraint over all regions of tissue using (7).

48. Solutions of the Poroelastic Equation of Motion
Sinusoidal steady-state
In the sinusoidal steady state, each of the variables, such as the displacement u(z,t), can be written in the form:
The equation of motion becomes:

49. Solutions of the Equation of Motion
For a sinusoidal displacement û0 that is applied to a sample of thickness , the boundary condition just under the platen is:
and the boundary condition at the bottom of the specimen is:
The solution to the equation of motion is:
where we have defined  to be the characteristic diffusion length:

50. Solutions of the Equation of Motion

51. Solutions of the Equation of Motion
Creep compression
In the case of a step application of stress 0 that is applied to a sample of thickness  and results in a creep compression, the initial condition is:
the boundary condition just under the platen is:
and the boundary condition at the bottom of the specimen is:

52. Solutions of the Equation of Motion
It turns out that the solution for the displacement of the surface just under the platen is:
where we have defined  to be the characteristic time constant:
Note that we can confirm the initial condition at t = 0+ by using the identity:

53. Solutions of the Equation of Motion
A curve-fit of the theoretical solution for the surface displacement of a cartilage specimen in confined compression from the linear KLM theory and the actual experimental data.

54. Physical Properties of Cartilage in Osteoarthritis
*DECREASED PROTEOGLYCN CONTENT
*INCREASED WATER CONTENT
=> explanation?
*DECREASED EQUILIBRIUM CONFINED COMPRESSION MODULUS
*INCREASED HYDRAULIC PERMEABILITY
*DECREASED STREAMING POTENTIAL

55. Cartilage: Summary of Key Points
Hyaline cartilage occurs at articular joints, fibrocartilage is found at other joints and locations
Long bone begins as cartilage
Articular cartilage bears frictional and compressive loads at joints
Cartilage is highly hydrated with few cells and no vessels; fibrillar collagen solid phase enmeshes charged proteoglycans and synovial fluid
Solid matrix elasticity, fluid pressure and viscosity, electrical charge density and osmotic chemical balance all contribute to viscoelastic cartilage mechanical properties
Osteoarthritis and rheumatoid arthritis lead to joint degeneration
Cartilage coefficient of friction is very low (~0.01)
Cartilage is often tested in compression
Poroelasticity theory models cartilage as a composite of elastic solid and viscous fluid